University of Groningen

The flapping flight of birdsThielicke, William

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Chapter IV

The Effects of Wing Twistin Slow-Speed Flapping Flight of Birds:Trading Brute Force Against Efficiency

Chapter IV

To be submitted to a scientific journalW.Thielicke and E.J. StamhuisThe effects of wing twist in slow-speed flapping flight of birds: Trading brute force againstefficiency

98

The effects of wing twist

ABSTRACT

Aircraft propellers are usually twisted, allowing the propeller blades to operate at a moreor less constant effective angle of attack over the full span. Twist enables the full blade tooperate at the angle of attack with the maximum lift to drag ratio, which enhances thepropulsive efficiency. Wing twist is sometimes also assumed to be essential in flappingflight, especially in bird flight. For small insects, it has however been shown that wingtwist has very little effect on the forces generated by a flapping wing. The unimportance oftwist was attributed to the prominent role of unsteady aerodynamic mechanisms. Thesewere recently also shown to play an important role in bird flight. It has therefore becomenecessary to verify whether wing twist is essential in the flapping flight of birds.

The aim of the study is to compare the efficiency and the aerodynamic forces of twistedand non twisted wings that mimick the slow-speed flapping flight of birds. The analyseswere performed with bird-like wing models that are equipped with different amountsof spanwise twist (0◦, 10◦, 40◦). The flow was mapped in three-dimensions around thewings using digital particle image velocimetry. The spanwise circulation and the induceddrag as well as the lift to drag ratio and the span efficiency were determined from thisdata.

The results show the development of leading-edge vortices (LEVs) on the non twistedwing at all flapping frequencies, and on the moderately twisted wing at the highestfrequency. LEVs did not develop on the highly twisted wing. Twisted wings were shownto generate significantly lower aerodynamic forces than wings without twist. However,twisted wings are more efficient. Efficiency and the magnitude of aerodynamic forces arecompeting parameters. Wing twist is hence beneficial only in the cases where efficiency isimportant – e. g. in cruising flight. Take-off, landing and manoeuvring however requirelarge and robust aerodynamic forces to be generated. This can be achieved by flapping thewings with only a small amount of twist which promotes the development of leading-edgevortices and increases the aerodynamic forces substantially. The additional force comesat the cost of efficiency, but it enables birds to perform extreme manoeuvres, increasingtheir overall fitness.

99

Chapter IV

INTRODUCTION

Wing twist is the torsion of a wing parallel to the spanwise axis, leading to a variationof the geometric angle of attack along span (see Figure 4.1). The propeller blades offixed-wing aircraft are typically twisted, decreasing the angle of attack at the tip of theblade and therefore compensating for the increasing circumferential velocities alongthe wing blade (Anderson, 2008). Twist allows the entire propeller blade to operate at amore or less constant effective angle of attack – close to the angle with the maximumlift to drag ratio (L/D, Walker et al., 2009). The individual propeller blade elementswill hence produce the least amount of drag for a given amount of lift. This means thaton a propeller, the torque is minimized for a given thrust. Hence the ratio of thrustproducing power to the mechanical power required to drive the propeller (propulsiveefficiency, Anderson, 2008) is increased by the application of twist. But not much canbe said about the magnitude of thrust producing power. In this context, the optimallyefficient propeller has a uniform inflow (and outflow) velocity over the whole propellerdisk, and each blade element operates at the effective angle of attack where profile draglosses are minimal (Gessow, 1948). These optimal effective angles of attack (in terms ofL/D) are typically in between 3 and 8 degrees for conventional airfoils, depending on theReynolds number (Re) and on the specific airfoil properties (Shyy et al., 2008).

Flapping wings are common in nature, and the aerodynamics of revolving propellersand flapping wings has some analogies. In fact, a revolving propeller approximates theaerodynamic situation of a flapping wing during the phase of the up- and downstrokein hovering flight (Usherwood & Ellington, 2002a). The analogy between revolvingand flapping wings is often used to explain why the wings of flapping-wing flyers haveto be twisted in the same tradition as aircraft propellers: At the wing tips, the lift of a

spanwise

chordwisevert

ical

spanwise

vert

ical

Fig. 4.1: Wing twist at mid-downstroke in an insect. Wing twist is the torsion of awing along the spanwise axis, leading to a variation of the geometric angle of attackover wing span. Left: Perspective view. Right: Frontal view.

100

The effects of wing twist

non-twisted, flapping wing is supposed to diminish due to stall because the effectiveangle of attack becomes too large (e. g. Herzog, 1968; Nachtigall, 1985). Stall can besuppressed by applying wing twist, and twist is supposed to enable the wings of birds,bats or insects to operate at their ’optimum’ (McGahan, 1973; Norberg, 1990) or ’mosteffective’ (Thomas & Hedenstroem, 1998) angle of attack, in analogy to aircraft propellers.It enables to maintain an ’appropriate’ (Alexander, 2004), ’favourable’ (Hubel, 2006) or’reasonable’ (Azuma, 2007) effective angle of attack at each wing section. In the flappingflight of insects, the analogy between flapping wings and twisted propellers has beenquestioned already, because the optimum, respectively the most effective angle of attackis not known for insects (Usherwood & Ellington, 2002a). Aerodynamic efficiency mightbe one important factor in natural flapping wing propulsion: Efficiency can be maximizedby adjusting the effective angle of attack towards the optimal L/D using twisted wings(e. g. Young et al., 2009; Walker et al., 2009). Measurements and simulations of modelwings mimicking hovering insect flight at low Re have, however, shown that wing twistdoes not measurably influence the overall L/D of the wings (Usherwood & Ellington,2002a; Du & Sun, 2008). In the flight of insects, it is likely that the generation of sufficientlifting force is more important than maximizing aerodynamic efficiency (Usherwood &Ellington, 2002a). Lifting forces can be maximized by operating wings at high effectiveangles of attack and generating stable leading-edge vortices (LEVs): LEVs enhance theaerodynamic force coefficients substantially, but are generally not associated with a highaerodynamic efficiency due to a significant increase of the drag component (e. g. Isogaiet al., 1999). LEVs are supposed to occur also in the flight of birds (Videler et al., 2004;Warrick et al., 2005; Hubel & Tropea, 2010; Thielicke et al., 2011; Muijres et al., 2012c;Chang et al., 2013, see Chapter III). Especially in slow-speed flight situations, duringmanoeuvring, take-off and landing, the enhanced force coefficients are required to enablethe generation of sufficient lifting forces under several physiological, anatomical andaerodynamic constraints (Lentink & Dickinson, 2009). In these situations, it is very likelythat the aerodynamic efficiency becomes of secondary interest – similar to insect flight.Studies on the effect of wing twist on the flow pattern in the slow speed flapping flight ofbirds have not yet been carried out. Therefore, the aim of this study is to analyse the effectof wing twist at Reynolds numbers and Strouhal numbers mimicking the slow-speedflight of birds. The focus of the present study is on the three-dimensional flow patternsthat are generated on and behind wings at several flapping frequencies and with differentamounts of twist. Furthermore, the aerodynamic efficiency and the circulation that canbe attained with twisted and non-twisted wings is analysed and the biological relevanceof the findings for the slow-speed flapping flight in birds is discussed.

101

Chapter IV

METHODS

wing modelling

Physical wing models with different amounts of twist are used to study the flow fieldin a water tunnel using digital particle image velocimetry (DPIV). The airfoil geometrydata used for modelling the wings were derived from measurements of a pigeon in freegliding flight (Biesel et al., 1985) and three-dimensional measurements of dissected wings(Bachmann, 2010, for more details, see Chapter III). The data were used to generateNACA 4-digit-modified-series airfoils (e. g. Ladson et al., 1996) for the wing models.The wings are equipped with a constant camber of 5% at 37% of the chord. Maximumthickness is located at 17% of the chord, the maximum thickness decreases linearly from10% (wing base) to 4% (wing tip). Additionally, the nose radius was modified with wingspan (base: 1; mid-wing: 0.5; wing tip: 0.1; where 1 denotes the radius equal to the originalnose radius, and 0 denotes a sharp leading-edge), as indicated by the airfoil geometrydata of the pigeon (Biesel et al., 1985; Bachmann, 2010). The single wing aspect ratio(AR = b/c, where wing span b = 120 mm and mean chord c = 43.75 mm) of the modelsis 2.74. The wings are mounted on a 3 mm steel rod, located at 30% of the chord. The wingbase is located 12 mm away from the two-degrees-of-freedom (2-DOF) joint, increasingthe effective wing span to 132 mm (see Figure 4.2). Three wing models with differentamounts of linear twist along the span were designed: The non twisted wing has 0◦twist,the moderately twisted wing is equipped with 10◦of twist, and the highly twisted wing isequipped with 40◦twist (see Figure 4.3).

The wings are equipped with a fixed amount of wing twist and do not adapt to changesin local velocities throughout the wing beat cycle. The results presented here can be seenas a first step and additional experiments with adaptive wing twist may have to follow infuture studies. The models were printed with a high resolution 3D printer (ZPrinter® Z310,layer thickness 0.1 mm, resolution 300 · 450 dpi, Z Corporation, Burlington, USA) andthe final wing models were casted with transparent epoxy resin (Epoxy casting resinwaterclear, Poxy-Systems® by R&G, Waldenbuch, Germany, refractive index 1.53). Dueto the refractive index being reasonably similar to water, flow measurements can beperformed in the direct vicinity of the flapping wings without shadows.

flow tank and kinematics

All measurement were performed in a recirculating water tunnel with transparent walls(test section = 250 · 250 · 500 mm, for more details see Chapter III), allowing to visualizethe flow from different views. The flow velocity was constant for all measurements

102

The effects of wing twist

Motor 1

Motor 2

2-DOF joint

Wing

z

y

x

Fig. 4.2: Flapping robot. Servomotor 1 drives the wing by means of an eccentric,servomotor 2 controls the geometric angle of attack.

base

mid

tip

c = 50 mm

c = 25 mm

span

= 1

20 m

m

0°

0°

0°

-5°

-10°

-20°

-40°

0° 0°

A B C D

Fig. 4.3: A: Planform of the wing models. B: non-twisted wing. C:moderately (10◦)twisted wing. D: highly (40◦) twisted wing.

103

Chapter IV

(Uf = 0.46 m/s). The wing was driven by a flapping mechanism that consists of twomechanically and electronically coupled servomotors (see Figure 4.2). The excursionangle of the wing and the geometric angle of attack were controlled throughout thewing beat cycle and synchronized trigger signals were sent to the high speed camera.The wing moves sinusoidally in a stroke plane set to 90◦with respect to the oncomingflow. The beat cycle starts with the upstroke, where the interaction of the wing with thefluid was minimized by adjusting the geometric angle of attack in order to minimize themean effective angle of attack of the wing (see Figure 4.4 for the definition of angles andvelocities on a flapping wing). The downstroke was performed with a constant geometricangle of attack (αgeo) of 0◦ ± 1◦at the wing base.

The Strouhal number St = fA/Uf determines the ratio between the flapping velocity,which is induced by the wing flapping at the frequency f with the amplitude A, and theforward velocity Uf. Three different Strouhal numbers (St = 0.2; 0.3; 0.4) are analysed,which are typically found in the flapping flight of birds (Taylor et al., 2003).

Wing twist alters the geometric angle of attack with wing span, and therefore adjuststhe effective angle of attack (αeff). The effective angle of attack for the different wingtypes and St during downstroke was determined using:

αeff(t, r) = αgeo(t, r) − αin(t, r) (4.1)

where t = time; r = radius of a wing element; αin = inflow angle, calculated as:

αin(t, r) = atan(rω(t)

Uf) (4.2)

where ω = angular velocity of the wing.The Reynolds number (Re = vtipc/ν, where ν is the kinematic viscosity) varied

slightly with St, and is in the range 2.2·104 < Re < 2.6·104.

flow field recording and analysis

The flow was visualized using polyamide tracer particles with 57 μm diameter (density =1016 kg/m3, Intelligent Laser Applications GmbH, Jülich, Germany) and a 5 W constantwave DPSS laser (Snoc electronics co., Ltd, Guangdong, China). Spherical and cylindricallenses were used to create a laser sheet with a thickness of about 1.5 mm. The flow wasfilmed using a high speed camera (A504k, Basler AG, Ahrensburg, Germany) set to aresolution of 1024·1024 pixels. Camera exposure was synchronized to the wing excursionwith an optomechanical trigger that initiated the exposure of the first image. The secondimage of the DPIV image pair was triggered with a custom delay system after exactly2 ms, which gave a mean particle displacement of 6 pixels. The particle density in theimages was 5.80 ± 0.48 particles per interrogation area (n = 7.4·103), and the particleimage diameter was 3.8 ± 1.6 pixels (n = 1.8·106) – conditions that are in the optimal

104

The effects of wing twist

Horizontal

Chord

Geo

metric

ang

le of attack

Inflo

w an

gle

Effective

angle of attack

Flap

pin

g v

elo

city

Forward velocity

Inflow velocity

Fig. 4.4: Definition of the velocity components and angles on a flapping wing atmid-downstroke.

range for PIV analyses (see Chapter II). The contrast in the images was enhanced priorto analysis using contrast limited adaptive histogram equalization (CLAHE, Pizer et al.,1987).

In most cases, the wing model appeared in the camera images. The position of thewing was extracted from all recordings to create a three-dimensional mask (see examplesin Figure 4.5) that was applied to the images before the analysis in order to prevent self-correlation (for more details, see Chapter III). A custom DPIV tool (PIVlab v1.31) wasused to derive velocities from the images. The tool uses an iterative multi-grid windowdeformation cross-correlation technique. Three passes with decreasing windows sizes(final window size = 34·34 pixels, with 50% overlap) were sufficient to generate precisevelocity maps in the two-dimensional test section (size = 160·160 mm, yielding 59·59vectors, vector spacing = 2.656 mm). The displacement map was validated and missingdata were interpolated (for more details see Chapter III).

Five successive downstrokes were recorded. PIV slices were captured from two direc-tions (see Figure 4.6), 59 positions with a distance of 2.656 mm were captured for eachthe vertical and the horizontal planes. Data acquisition at different planes was enabledwithout the need for re-calibration by displacing the camera and the laser sheet at thesame time (for more details see Chapter III). Due to the highly periodic nature of theflow, the planes could be captured at separate stroke cycles. The combination of thevelocity data gives a three-dimensional representation of the flow in a test volume of160·160·160 mm around the wing. The resulting Cartesian grid (59·59·59 points) containsthe full three-dimensional velocity information at each point.

Vortices were visualized with iso-surfaces of the positive second invariant Q of thevelocity gradient tensor, a scalar quantity that reliably detects vortical regions without

105

Chapter IV

Wing beat cycle [%]6 14 23 31 40 57 68 82

xz

y

Fig. 4.5:Wing positions as they were extracted from the PIV recordings to generate3D masks. Yellow: Data from the xz plane. Blue: Data from xy plane.

A B

Flow Flow

x

z

y

Fig. 4.6: Several cross-sections through the test volume were captured from twodirections. A: 59 sections in the xz plane capture u and w velocity components. B:59 sections in the xy plane capture u and v velocity components. Data from bothdirections were combined, yielding a three-dimensional representation of the flowfield. The black arrow indicates the direction of the oncoming flow, the red arrowindicates the stroke plane.

106

The effects of wing twist

being prone to shear (e. g. Hunt et al., 1988, for more details see Chapter III). Vortices arepresent if streamlines or a texture generated via line integral convolution (LIC, Cabral& Leedom, 1993, which is functionally equivalent) circle around a focus when viewedfrom a frame of reference moving with the vortex (Robinson et al., 1989). The focus mustcoincide with a broad peak in vorticity and Q. This vortex is defined as leading-edgevortex, if it is located on top of the wing and close to the leading-edge and if a regionwith reversed flow exists on top of the wing.

The circulation along the spanwise axis of the wings was calculated by integratingspanwise vorticity in the xy-plane for each wing section. The results were very consistentcompared to an alternative approach, the integral of tangential velocity along a looparound the wing in the xy-plane (for more details see Chapter III). The approach ofBirch et al. (2004) to derive sectional lift is followed, which is based on the circulationtheorem. This theorem is normally appropriate only for steady flow conditions in two-dimensional flows, but has been shown to give reliable results for similarly unsteady flowsat comparable Re (Unal et al., 1997). The sectional circulatory lift at mid-downstrokeL ′circ is calculated from the product of fluid density, free flow velocity and local spanwise

circulation:L ′circ(z) = ρUfΓ(z) (4.3)

where ρ = density, z = spanwise position, Γ(z) = spanwise circulation at mid-downstroke.Integrating L ′

circ over wing span gives the total circulatory lift (Lcirc). As only span-wise circulation is included in this lift estimate, the real lift of the wings will be un-derestimated (Birch et al., 2004; Poelma et al., 2006). Due to the identical planform,airfoils, kinematics and experimental conditions of the wing types that are tested, therelative errors are expected to be constant. Hence, the results are nondimensionalizedwith respect to Lcirc of the ’standard experiment’: the non twisted wing at St = 0.3. Theinduced drag (drag due to lift, Anderson, 2007) was estimated by assuming a momentumbalance upstream and downstream of the flapping wing (e. g. McAlister et al., 1995; Giles& Cummings, 1999):

Dind =12ρ

∫A

((v2down +w2down) − (v2up +w2

up))dA (4.4)

where vup respectively wup represent the vertical respectively spanwise velocities up-stream of the wing and vdown respectively wdown represent the velocities downstreamof the wing in the yz-plane.

The results (again nondimensionalized with respect to the non twisted wing at St= 0.3) were used to calculate the ratio of circulatory lift to induced drag (Lcirc/Dind).Due to the nondimensionalization, the non twisted wing has a Lcirc/Dind of unity. Thisratio can be interpreted as a relative measure for aerodynamic efficiency, analogous tothe L/D of fixed wings. Note that profile drag and additional sources of lift are ignored inLcirc/Dind.

107

Chapter IV

Another common measure for aerodynamic efficiency, that has also been applied toflapping flight of insects (e. g. Bomphrey et al., 2006), bats (Muijres et al., 2011, 2012b)and birds (Muijres et al., 2012b), is the span efficiency (ei, for details, see Bomphrey et al.,2006; Henningsson & Bomphrey, 2011):

ei =4

πb2

(∫b/2−b/2 vdown(z)

√b2 − 4z2dz)2∫b/2

−b/2 v2down(z)

√b2 − 4z2dz

(4.5)

where b = wing span.The span efficiency relates the ideal induced power required to generate a certain

amount of lift to the real induced power that is required. The ’ideal wing’ (with anelliptic distribution of circulation and a uniform downwash behind the wing) requiresthe minimum possible induced power (Bomphrey et al., 2006), and has a span efficiencyof unity. Any deviation from the uniform downwash will increase the induced power,and therefore decrease span efficiency.

Statistical tests for the equality of means are conducted following the recommendationsin Lozän & Kausch (1998) and Kesel et al. (1999) with a significance level of 5%. A Lillieforstest is used for testing normal distribution.

108

The effects of wing twist

RESULTS

The effective angle of attack of the three different wings during downstroke was deter-mined for the three Strouhal numbers tested using basic trigonometry. In most cases, thetheoretical effective angle of attack peaks at the wing tip at mid-downstroke (see Figure4.7). The non twisted wing experiences the highest effective angles of attack and also thehighest gradients. In the twisted wings, the peak effective angles of attack are reduced by10◦respectively 40◦. Wing twist ’overcompensates’ the inflow angle in the highly twistedwing at a Strouhal number of 0.2, resulting in a negative effective angle of attack at thewing tip (see Figure 4.7).

The 3D flow field is captured by recording 2D slices from two different directions.These slices cannot be captured at the same time, which is a potential source of error ifthe flow is not perfectly periodic. However, taking a phase average of five frames doesnot substantially alter qualitatively or quantitatively, but it does slightly reduce noise (seeFigure 4.8). Therefore, all the following measurements and figures are the mean ± s.d. offive measurements.

First, two-dimensional cross-sections are checked for the existence of vortices. Thecross-sections in the xy-plane at 2/3 span reveal the existence of leading-edge vorticeson some of the wings (see Figure 4.9). Both the magnitude of vorticity (see colour map)and the induced flow velocities (see vector scale) increase with St and decrease withtwist. The non twisted wing creates LEVs at all St: At St = 0.2, the LEV is small andvery close to the wing surface, but increases in size at St = 0.3. At St = 0.4, the LEV hasgrown remarkably and shifts away from the wing substantially, indicating large scaleflow separation. The moderately twisted wing generates a LEV only at St = 0.4. At thisStrouhal number, the centre of the vortex is located on top of the wing, indicating a region

0 50 100

0

10

20

30

40

50

Wing span [%]

� eff [°

]

0 50 100

0

10

20

30

40

50

Wing span [%]0 50 100

0

10

20

30

40

50

Wing span [%]

no twist 10° twist 40° twist

St = 0.2 St = 0.3 St = 0.4

� eff [°

]

� eff [°

]

Fig. 4.7: Effective angle of attack as a function of wing span at mid-downstroke.Both wing twist and St determine the spanwise distribution of the effective angleof attack.

109

Chapter IV

Five frame average

u v

elo

city

v ve

loci

tyw

vel

oci

tyvo

rtic

ity

z

Instantaneous

0.7

u [m/s]

0.2

0.2

v [m/s]

w [m/s]

� [1/s]

-0.3

0.15

-0.15

120

-120

Fig. 4.8: The effect of phase averaging. The fundamental flow patterns are notaffected, butmeasurement noise slightly decreases. Non twistedwing, St= 0.3, 66%span, mid-downstroke.

110

The effects of wing twist

no

tw

ist

10° t

wis

t40

° tw

ist

St = 0.4St = 0.3St = 0.2

0.44 m/s −100 −50 0 50 100

�z

Fig. 4.9: 2D cross-section at 2/3 span atmid-downstroke. Themagnitude of vorticity,aswell as the flow velocities increasewith St anddecreasewithwing twist. Spanwisevorticity is shown in colour. The texture is generated using line integral convolution.The inflow velocity is subtracted from the vector map.

of recirculating flow. At lower St, there is no recirculating flow on top of the moderatelytwisted wing. The highly twisted wing does not create leading-edge vortices at any St

and the interaction with the fluid is generally very small. A significant circulation of fluidaround the wing is hardly generated when St � 0.3.

The three-dimensional analyses provide further insight into the detailed nature ofthe flow field: Visualizations of the Q-criterion reveal the shape of the vortex system(see Figure 4.10). In the non twisted wing, the LEV increases in size towards the wingtip and merges with the tip vortex. At St = 0.4, the LEV becomes relatively unstable,which is indicated by several vortical structures that separate from the wing. It appearsthat a LEV is also present on the moderately twisted wing at St = 0.3. But the 2D resultspresented earlier (see Figure 4.9) have shown that this is not the case, as this vortexfails some of the criteria for a LEV (no recirculating fluid on top of the wing). At St =0.4 however, the moderately twisted wing creates a stable LEV. The highly twisted wingseems to generate only very weak vortices that do hardly appear in the visualization withthe selected threshold for the Q-criterion. The tip vortex – which is a good indicator forthe generation of lift on finite wings – is too weak to appear in the visualization exceptfor the highest Strouhal number.

The strong influence of St and twist on the flow patterns is also demonstrated in thevisualization of the 3D downwash distribution (see Figure 4.11): In most cases, significantdownwash is generated over a large part of the span (the visualization shows isosurfacesfor downwash velocities > 0.2 · Uf). Peak downwash velocities are located close to the

111

Chapter IV

St = 0.2n

o t

wis

t10

° tw

ist

40° t

wis

tSt = 0.3 St = 0.4

Fig. 4.10: Three-dimensional visualization of the flow at mid-downstroke. Iso-surfaces of the Q-criterion (Q > 1200). Leading-edge vortices appear on the nontwisted wing, and on the moderately twisted wing if St = 0.4. The highly twistedwing generates much weaker vortices that hardly appear in the visualization.

inner boundary of the tip vortex. The volume of fluid that is imparted with a significantdownwash velocity component becomes smaller in the twisted wings due to the smalleffective angle of attack. As already shown in the visualization of the Q-criterion, thehighly twisted wing has the least amount of interaction with the fluid. Only at the highestSt, a large volume with downwash velocities > 0.2 · Uf becomes visible. In summary,the volume of fluid with considerable downwash increases with St, and decreases whenwing twist is applied.

Spanwise flow in the core of the leading-edge vortex is a feature described in severalstudies that analyse or simulate flapping wings. Significant spanwise flow in the coreof the leading-edge vortices can not be found. Instead, some weak positive (from baseto tip) spanwise flow on top of the wing and behind the LEV is found (see Figure 4.12).In many cases, there is a positive spanwise flow component close to the wing tip whichis caused by the rotation of the tip vortex. A negative spanwise flow component (from

112

The effects of wing twist

St = 0.2n

o t

wis

t10

° tw

ist

40° t

wis

tSt = 0.3 St = 0.4

Fig. 4.11: Downwash at mid-downstroke for the different wings and St tested. Thethreshold for the iso-surfaces is 0.2 · Uf. The downwash is considerably reduced inthe twisted wings.

tip to base) is generated on the top side of the tip vortex. This negative spanwise flowcomponent sometimes extends over the outer half of the wing.

The lift generated by bound vortices (’conventional’ bound vortex and leading-edgevortex) is determined by the total bound circulation of the wing. All wing types create apositive circulation at mid-downstroke at all St under test (see Figure 4.13). This mightbe surprising, as the wing with 40◦twist is operating at a slightly negative effective angleof attack at St = 0.2 (see Figure 4.7). However, the zero-lift angle of attack for the testedwing is about -3◦, which explains the generation of positive circulation at slightly negativeeffective angles of attack. The circulation increases considerably towards the wing tip inmost cases (see Figure 4.13). Circulation also increases with St, but decreases stronglywhen twist is applied. An elliptic distribution of circulation over span is desirable tominimize the induced drag for steadily translating, fixed wings. The circulation of theflapping wings departs remarkably from the theoretically optimal elliptic distribution inmost cases. Only the highly twisted wing at St � 0.3 shows a distribution of circulationthat is comparable to the elliptic distribution (see Figure 4.13). In Figure 4.14, the relative

113

Chapter IV

St = 0.2n

o t

wis

t10

° tw

ist

40° t

wis

tSt = 0.3 St = 0.4

Fig. 4.12: Spanwise flow of the different wing types at all St tested at mid-downstroke. Red: Flow from tip to base exceeding 0.1 ·Uf. Green: Flow frombase totip exceeding 0.2 · Uf. Regions with elevated positive and negative spanwise flowcomponents are mainly linked to the rotation of the tip vortex. Only a weak flowfrom base to tip is present on top of the wings that generate leading-edge vortices.

difference of measured versus elliptic distribution of circulation is plotted over span. Anydeviation from zero indicates a deviation from the elliptic distribution. The smallestdeviation is found in the highly twisted wing where also the gradient in the effectiveangle of attack is weakest (see Figure 4.7). Here, the relative deviation from the ellipticdistribution increases slightly with St. Both the non twisted and the moderately twistedwing have a comparable relative deviation from the elliptic distribution of circulation(see Figure 4.14). Because the relative difference is comparable, the absolute differenceincreases with St and decreases with wing twist.

Deviations from the elliptic distribution of circulation will increase the induced dragof the flapping wing. The induced drag was calculated from the yz planes at several xpositions using Equation 4.4. In preliminary calculations, Dind was found to be maximalat mid downstroke at the trailing edge of the wing.

114

The effects of wing twist

0 50 100 0

0.0020.0040.0060.008 0.01

0.0120.0140.016

Γ z [m

2/s

]

Wing span [%]0 50 100

0

0.005

0.01

0.015

0.02

0 50 100 0

0.005

0.01

0.015

0.02

0.025

0.03

no twist 10° twist 40° twist

St = 0.2 St = 0.3 St = 0.4

Fig. 4.13: Spanwise circulation along span atmid-downstroke for the different wingtypes and St tested. The circulation increases with St. Wing twist leads to a declineof circulation. The deviation froman elliptic distribution is in some cases remarkable.

no twist 10° twist 40° twist

0 50 100−0.8−0.6−0.4−0.2

0 0.2 0.4 0.6 0.8

ΔΓz

Wing span [%]0 50 100

−0.8−0.6−0.4−0.2

0 0.2 0.4 0.6 0.8

0 50 100−0.8−0.6−0.4−0.2

0 0.2 0.4 0.6 0.8St = 0.2 St = 0.3 St = 0.4

Fig. 4.14: Relative deviation of themeasured circulation from an elliptic distributionof circulation with equal mean circulation. Expressed as a fraction of the meancirculation. The highly twisted wing shows the smallest deviation, whereas the nontwisted and the moderately twisted wing both show a similar performance.

In all wings, the nondimensionalized Lcirc and Dind increase with St. But thereare considerable differences between the lift and drag created by wings with differentamounts of wing twist (see Figure 4.15). The non twisted wing generates the highest forces,followed by the moderately twisted wing. The offset between the lift forces generated bydifferent wing types is very constant. This is not the case for the drag forces. Here, thenon twisted wing generates an exceptionally high drag at increasing St. The lowest liftand drag are generated by the highly twisted wing (see Figure 4.15): Compared to thenon twisted wing, the highly twisted wing generates between 27.1 - 49.4% of circulatorylift and between 6.3 - 19.7% of induced drag.

Plotting the nondimensionalized data over the mean effective angle of attack (αeff) atmid-downstroke shows thatLcirc andDind can be modelled withLcirc = sin(αeff)cos(αeff)

respectively Dind = sin2(αeff) (Dickson & Dickinson, 2004, see Figure 4.16). The agree-ment between the experimental data and the calculated fit is reasonable and does notdepend on the amount of twist of the wing.

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Chapter IV

0.2 0.3 0.4 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

No

rmal

ized

Din

d

Strouhal nr.0.2 0.3 0.4

0

0.2

0.4

0.6

0.8

1

1.2

No

rmal

ized

Lci

rc

Strouhal nr.

no twist 10° twist 40° twist

Fig. 4.15: Nondimensionalized lift and drag. Both increase with St. Wing twistreduces lift by a factor of up to 2.5 and drag by a factor of up to 6.7. Due to thenondimensionalization, the non twisted wing at St = 0.3 has a Lcirc respectivelyDind of unity. All means are significantly different (α = 0.05).

0 10 20 300

0.5

1

1.5

mean αeff

[°]

No

rnam

ilzed

Lci

rc

0 10 20 300

0.5

1

1.5

mean αeff

[°]

No

rmal

ized

Din

d

A B

Fig. 4.16: Nondimensionalized lift and drag vs. the mean effective angle of attackat mid-downstroke. A: Normalized Lcirc. B: NormalizedDind. Red = no twist, green= moderate twist, blue = high twist. Due to the nondimensionalization, the nontwisted wing at St = 0.3 has a Lcirc respectivelyDind of unity. Solid lines representleast-squares fits: Lcirc = sin(αeff)cos(αeff)n+ l0n = 1.895; l0 = 0.1751; R2 = 0.89;Dind = sin2(αeff)n+ d0n = 4.828; d0 = -0.01765; R2 = 0.93n accounts for the nondimensionalization, and l0 respectively d0 account for thenon-zero force at zero degrees effective angle of attack of the wings.

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The effects of wing twist

0°

St = 0.2 St = 0.3 St = 0.4

10°

40° 0°

10°

40° 0°

10°

40°0

21

4

6

8

3

5

7

no

rmal

ized

L

/D

ind

circ

Fig. 4.17: Lcirc/Dind for the differentwing types. The highest ratio is achievedwith highly twisted wings at low St. Anyincrease in the effective angle of attackvia twist or St reduces Lcirc/Dind. Allmeans are significantly different (α = 0.05).Dueto thenondimensionalization, thenontwisted wing at St = 0.3 has a Lcirc/Dind

of unity.

0 10 20 300

2

4

6

8

mean �eff

[°]

L circ

/Din

d

Fig. 4.18: Lcirc/Dind vs. effective angleof attack at mid-downstroke. Data from alltwist angles and St is pooled. The solid linerepresents a least squares fit of the functionLcirc/Dind = 1/tan(αeff + k) · nk = 0.1502; n = 0.8733; R2 = 0.9813n accounts for the nondimensionalization.k accounts for the fact that Lcirc/Dind ispositive for negative effective angles of at-tack (due to the zero-lift anglebeing smallerthan zero), and 1/tan(0) not being defined.

The dissimilar relation of lift respectively drag to St and wing twist and has a stronginfluence on Lcirc/Dind (see Figure 4.17): The highly twisted wing has the highestLcirc/Dind when compared to the other wing types at the same Strouhal number. Theratio increases with twist. Additionally, there is also a strong dependence on St. Anincrease in St leads to a decrease of Lcirc/Dind (see Figure 4.17). The mean effectiveangle of attack αeff on the wing at mid-downstroke is positively related to St and neg-atively related to wing twist (see Figure 4.7). Figure 4.18 shows the relation betweenLcirc/Dind and αeff including all St. Lcirc/Dind decreases substantially when αeff atmid-downstroke increases. This trend can reasonably be modelled usingLcirc/Dind = cos(αeff)/sin(αeff) = 1/tan(αeff) (see Figure 4.18).

The superior aerodynamic efficiency of wings that are operating at low St and thatare equipped with twist has been demonstrated by the measurements of the circulationdistribution and by Lcirc/Dind. Further support for the increasing efficiency is derivedfrom the distribution of downwash velocities along span: The optimal wing with anelliptic distribution of circulation will induce a constant downwash velocity along thespan (Anderson, 2008). Any deviation from uniformity decreases efficiency. Such auniform downwash distribution can only be observed for the highly twisted wing at St� 0.3 (see Figure 4.19). This is in good agreement with the nearly elliptic distribution of

117

Chapter IV

20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

Do

wn

was

h [m

/s]

Wing span [%]20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5St = 0.2 St = 0.3 St = 0.4

no twist 10° twist 40° twist

Fig. 4.19: Downwash distribution over span at mid-downstroke directly behindthe trailing edge.The highly twisted wing creates the most uniform downwash,followed by the moderately twisted wing. The non twisted wing displays the mostunfavourable downwash distribution with the highest gradients from base to tip.

0°

St = 0.2 St = 0.3 St = 0.4

10°

40° 0°

10°

40° 0°

10°

40°

0.85

0.9

0.95

1

Span

eff

icie

ncy

1/k

ind

n.s.:

Fig. 4.20: Span efficiency of the wings atmid-downstroke. Span efficiency is gen-erally high, and confirms the trend thatwas found in Lcirc/Dind. Non-significantdifferences (n.s.) are highlighted (α = 0.05).

spanwise circulation (see Figure 4.13 & 4.14). Both the non twisted and the moderatelytwisted wing deviate largely from the uniform downwash distribution (see Figure 4.19).The deviation grows considerably with St, and the non twisted wing always generatesthe most unfavourable downwash distribution. Due to large scale flow separation at St =0.4 (see Figure 4.10), a double peak in the downwash velocity can be observed. Thesequalitative insights on the downwash distribution can be further specified by comparingspan efficiency. Due to some noise in the flow velocities directly behind the trailing edgeof the wing (caused by the rolling-up of the boundary layer), the results are less clear thanthe results of Lcirc/Dind (which are based on integral quantities), but show very similartrends (see Figure 4.20): The span efficiency increases when the wings are progressivelytwisted. The highly twisted wing has a span efficiency that is very close to unity. There isno clear trend for the dependency of span efficiency vs. St.

118

The effects of wing twist

DISCUSSION

flow patterns

Wing twist and Strouhal number considerably influence the flow patterns generated byflapping wings. Vortex growth is related to the rate of change in circulation, which is inturn determined by the effective angle of attack (Nudds et al., 2004). The effective angleof attack is related to the Strouhal number which expresses the ratio of Uf and flappingvelocity. Wing twist is the ’counterpart’ to St: It lowers both the gradient along span aswell as the magnitude of the effective angle of attack. Leading-edge vortices develop ifthe effective angle of attack is exceeding a certain threshold. Vorticity accumulates in aLEV over time, supposed that there is no effective vorticity drain (Bomphrey et al., 2005).Previous studies (e. g. Ellington et al., 1996; Willmott et al., 1997; van den Berg & Ellington,1997) found high spanwise flow components in the LEV which are supposed to drainvorticity away from the wing into the tip vortex and to play a key role in LEV stability.In the present study, the LEVs are stable throughout the duration of the downstroke inmost cases, significant spanwise flow in the vortex core itself does not develop, regardlessof wing twist. Weak spanwise flow was found only behind the LEV on top of the wing.Here, the fluid is presumably accelerated radially by ’centrifugal pumping’ (Lentink& Dickinson, 2009). The absence of an effective vorticity drain most likely limits themaximum acceptable rate of vorticity accumulation, which is demonstrated by the largescale flow separation in the non-twisted wing at the highest Strouhal number. We havespeculated earlier, that the low velocity gradient over the span of a flapping wing intranslation hinders the development of a significant pressure gradient that could drivespanwise flow (see Chapter III). The additional application of wing twist reduces thegradient in angle of attack along wing span even more and further reduces the pressuregradients. It therefore appears plausible that spanwise flow is minimal on twisted wingsin translational flow. Wing twist also reduces the effective angle of attack, greatly loweringthe rate of vorticity accumulation in the LEV. An effective vorticity drain is therefore notessential for vortex stability.

circulation and force

The application of wing twist greatly reduces the amount of total bound circulation(proportional to lift) on the wing. In a study on revolving wings at lower Re (Re =8000, Usherwood & Ellington, 2002a), it was shown that the presence of wing twist doesnot result in different polar diagrams (lift plotted over drag). Altering the amount ofwing twist had the same effect as altering the geometric angle of attack of the wing base

119

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(Usherwood & Ellington, 2002a). Thus, lift was shown to be proportional to the effectiveangle of attack of the wing, no matter what the twist angle was. Unlike the study fromUsherwood & Ellington (2002a), the geometric angle of attack at mid-downstroke wasnot altered in the present study. Instead, the effective angle of attack was altered using twodifferent means: Strouhal number and wing twist. Wings that are flapping at comparableeffective angles of attack should generate comparable forces according to the results fromUsherwood & Ellington (2002a) on revolving wings: In the present study, the effectiveangle of attack of the moderately twisted wing at St = 0.3 is very similar to the effectiveangle of attack of the non twisted wing at St = 0.2 at mid-downstroke (see Figure 4.7).Furthermore, the effective angle of attack of the moderately twisted wing at St = 0.4and the non twisted wing at St = 0.3 are very comparable (see Figure 4.7). Despite thesesimilarities, the measurements show that the forces are slightly different (see Figure 4.15).One very likely reason for these different forces is the different size of the LEVs (seeFigures 4.9 & 4.10). As mentioned earlier, one parameter that determines the size of aLEV is the time span over which vorticity may accumulate in the vortex. Higher flappingfrequencies result in less time for vorticity accumulation and explain the slightly differentsize of the LEV despite comparable effective angles of attack. Therefore, the results of thepresent study are in agreement with Usherwood & Ellington (2002a): Also in flappingwings at higher Re, the effective angle of attack is the main parameter responsible for themagnitude of aerodynamic forces, together with the duration of the downstroke. Wingtwist per se is of minor importance for the aerodynamic forces. Further support for thisconclusion comes from the trigonometric relation of αeff and Lcirc respectively Dind,that holds for all wing types under test (see Figure 4.16). This relation has been foundpreviously in studies on hovering insects (Dickinson et al., 1999; Usherwood & Ellington,2002a) and also in a study that included forward flight of insects at very low Re (Dickson& Dickinson, 2004). The present study shows that the trigonometric relation may alsobe applied to flapping wings at higher Re.

Lcirc increases even if the local effective angle of attack exceeds the stall angle ofsteadily translating wings (between 8◦and 15◦, Anderson, 2007). There is no suddenchange in forces with the onset of leading-edge vortices. The non-twisted wing has thepotential to create much larger lift and drag – simply due to the larger effective angle ofattack. Lcirc and Dind however scale differently, the drag component increases relativelymore than the lift component, and this will influence the efficiency.

efficiency

Two measures for quantifying efficiency are used. In addition to the mechanical flightefficiency (related to Lcirc/Dind), the efficiency of lift generation was measured (relatedto the span efficiency respectively the distribution of circulation). These two independentparameters (Muijres et al., 2012b) both increase substantially with wing twist. The morethan 4-fold difference in Lcirc/Dind between twisted and non-twisted wings (see Figure

120

The effects of wing twist

4.17) is most likely an overestimation, as other (constant) sources for drag were ignored –these will attenuate the relative differences. It is known that the generation of aerodynamicforces under the presence of leading-edge vortices reduces the mechanical flight efficiency(e. g. Lentink & Dickinson, 2009), and that operating a flapping wing at an αeff just belowthe limit of leading-edge separation enhances efficiency (Culbreth et al., 2011). A LEVincreases the total aerodynamic force and the gain in lift is accompanied by increaseddrag due to the loss of the leading-edge-suction force (Polhamus, 1966). Delta-wingaircraft at high Re, but also revolving wings with different amounts of twist at very lowRe that generate lift via the LEV, were shown to have a L/D that is inversely proportionalto tan(α) (Polhamus, 1971; Usherwood & Ellington, 2002a; Altshuler et al., 2004). Asthe results of the present study show, this relation also holds for flapping wings at higherRe mimicking the slow speed flight of birds. In the flapping wings that were tested, anyforce enhancement that is caused by an increase in effective angle of attack comes at thecost of reduced efficiency. This is not fundamentally different from a finite wing in purelysteady conditions (see Figure 4.21). Here, the peak of the maximum force is found atα ≈ 16◦, just before the wing stalls. The highest efficiency (in terms of L/D) is foundat smaller angles of attack however (note that in Figure 4.21B, the drag at zero degreesangle of attack was subtracted from the drag measurements. In reality, the optimumL/D will shift towards slightly higher α). Efficiency and maximum aerodynamic forcehence are competing parameters also under steady flow conditions: Airplanes cannotfly at the maximum L/D in situations that require large forces, like take-off and landing,because efficiency and force coefficients cannot be maximized simultaneously (Anderson& Eberhardt, 2001; Anderson, 2008). In flapping wings, the peak total force coefficient isgenerated at very high effective angles of attack, because the wing does not stall in theconventional sense. Maximum efficiency and maximum total force are therefore found atvery opposed effective angles of attack, and seem to be even more competing parametersthan in steady flow conditions.

The efficiency of lift generation was further analysed with two closely coupled measures– the spanwise distribution of circulation and the spanwise distribution of downwash.The latter was used to calculate the span efficiency – a measure for the efficiency of liftgeneration (Muijres et al., 2012b). The best agreement between elliptic distribution ofcirculation and the measured circulation was found in the highly twisted wing at low St –a situation where the interaction with the fluid is small and only little lift is generated.Spanwise circulation is positively related to the effective angle of attack (Nudds et al., 2004)and velocity. Both parameters increase with span on a non twisted, flapping wing andpotentially yield a distribution of circulation that deviates from the elliptic distribution.To compensate for the increasing effective angle of attack along span, a wing could beequipped with twist, eventually making the effective angle of attack constant along thewing. Even with such a constant effective angle of attack, the velocity gradient alongspan will still yield a distribution of circulation that is not elliptic. If other parameters areconstant, this could only be compensated for by further decreasing the effective angle of

121

Chapter IV

A: Flapping B: Steady

Ftot

Ftot

LcircDind

LD-d0

0 10 20 30−0.2

0

0.2

0.4

0.6

0.8

1

Angle of attack [°]

No

rmal

ized

L/D

an

d F

tot

0 10 20 30−0.2

0

0.2

0.4

0.6

0.8

1

Mean effective angle of attack [°]

No

rmal

ized

Lci

rc/D

ind a

nd

Fto

tEfficiency

Force +

-+

-

Efficiency

Force +

-+

-

Fig. 4.21: A: Total aerodynamic force Ftot = (L2circ+D2ind)

0.5 (blue) and Lcirc/Dind

vs. the mean effective angle of attack. All twist angles and St are pooled. Thetotal force increases with αeff and the efficiency decreases with αeff. B: Totalaerodynamic force Ftot = (L2 + D2)0.5 and L/(D − d0) of the non twisted wingunder fully steady conditions, as measured in a wind tunnel. L/(D − d0) followsa similar trend as in the measurements of the flapping wings. The increase of Ftotstops at 16◦(stall). Each parameter is normalized with respect to its maximum valuein the measurement.

attack with span by additional twist. This is the case in the highly twisted wing at thelowest Strouhal number: The effective angle of attack at the wing tip is smaller than atthe base – it compensates for the higher flow velocities at the wing tip. Subsequently, thedistribution of circulation is elliptic, but the circulation and the resulting lift are almostnegligible, as lift scales with αeff. From this perspective, it appears questionable whetheran elliptic distribution of circulation can be desirable on a flapping wing if it is supposedto generate significant lift. The results of the downwash distribution and span efficiencysupport these conclusions. Span efficiency increases with wing twist, as the gradientin effective angle of attack diminishes and the downwash distribution becomes moreeven as a consequence. Despite the large variation of twist and St that was tested in thepresent study, the range of span efficiencies appears to be relatively small: The lowestspan efficiency is 83.6 ± 0.8% and the highest span efficiency is 98.6 ± 0.4%. This iscomparable to the span efficiencies reported for the flapping flight of several bird species(86% to 95%, Muijres et al., 2012b), indicating that the effective angle of attack in birdsmight vary similarly as in the present study. It has to be kept in mind that span efficiencyis inherently sensitive to noise in the downwash measurements and any irregularities inthe downwash distribution. A comparison of the result with other measurements thatwere taken under different circumstances and with different methods should thereforeonly be made with caution. The measurements of the present study however support the

122

The effects of wing twist

idea that twisted wings have the potential to generate lift more efficiently – in addition tothe increase in efficiency caused by the higher Lcirc/Dind.

twist in nature’s flapping wing flyers

Desert locusts are supposed to benefit from the efficiency of attached flow aerodynamicsat the cost of reduced peak aerodynamic forces by using twisted wings (Young et al., 2009).These organisms are highly migratory, and they might not need the extra forces associatedwith LEVs that appear at elevated αeff, as they are supposed to be less manoeuvrablethan other insect species (Bomphrey, 2012). Eventually, the ability of desert locuststo accelerate very rapidly from rest to the minimum flight speed via jumping (Katz &Gosline, 1993) renders the requirement of generating large forces at low flight velocitiesless important, and allows to use wing twist to increase efficiency. Butterflies are anotherexample where wing twist enhances aerodynamic efficiency, but also reduces peak liftforces during the downstroke (Zheng et al., 2013b).

In the cruising flight of birds, peak lift forces are most likely not of primary importance.Here, energetic efficiency is likely to play a major role due to the high energetic costs andthe long duration of cruising flight periods (e. g. Norberg, 1990). Birds can afford to avoidthe high drag that would come with the development of leading-edge vortices (Nuddset al., 2004; Park et al., 2012): The application of wing twist helps to find the optimumbalance between aerodynamic efficiency and the required aerodynamic forces duringcruising. Cruising flight with a close-to-optimal L/D therefore seems to be possible.Furthermore, the gradients in velocity and αeff over wing span are inherently weakerin cruising flight than in slow speed flight. Airfoil shape, wing planform and twist cancompensate for some of the gradients in circulation over wing span (e. g. Anderson,2008). Bird wings are cambered at the wing base and more flat close to the tip (e. g.Nachtigall & Wieser, 1966; Liu et al., 2004). Wing camber increases lift with attached flowaerodynamics (e. g. Okamoto et al., 1996, Chapter III of this thesis), and the spanwisedistribution of camber in combination with twist could be a strategy to increase the spanefficiency in cruising flight.

The story looks however different in slow speed flight, during manoeuvring, take-offand landing: The selection pressure to avoid being killed by predators is very high inbirds: The ability to take-off rapidly and to manoeuvre quickly will decrease the chanceof a bird to be killed (e. g. Lima & Dill, 1990; Swaddle & Lockwood, 1998; van den Houtet al., 2010). In predator escape, rapid accelerations require large forces to be generated bythe wings. Aerodynamic efficiency does not seem to be an important target of selectionin these situations (Curet et al., 2013). According to the results of the present study, wingtwist is very disadvantageous when such large forces are required. Furthermore, theability to fly very slowly just before landing will reduce the chance of injury or wingdamage. Keeping the wings perfectly intact is important, as the flight performance duringtake-off, manoeuvring and escape reactions decreases substantially with damaged wings

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Chapter IV

(e. g. Tucker, 1991; Swaddle & Witter, 1997; Chai et al., 1999). As stroke amplitude andflapping frequency in birds are constrained (Lentink & Dickinson, 2009), slow flightrequires high lift coefficients. These can best be achieved by operating the wings at highangles of attack. Stall does not seem to be a primary issue on flapping or revolving wings(Usherwood & Ellington, 2002b; Thielicke et al., 2011; Ozen & Rockwell, 2012, ChapterIII of this thesis), and lift continues to increase until very high effective angles of attackunder the presence of leading-edge vortices. LEVs increase lift and drag at the same timeand enable manoeuvres that are essential for bird flight. Despite the implication of theword, the increase in drag does not always need to be disadvantageous. Lift and dragboth contribute to the total aerodynamic force. If the stroke plane is set correctly, all ofthe total aerodynamic force can be used to offset weight. This has been shown previouslyfor the flapping flight of dragonflies: Drag can be used to support three quarters of theweight, and potentially, the required power for flight can be reduced by a factor of two(Wang, 2004). As the results of the present study have shown, this might for a goodpart also be applicable to the slow-speed flapping flight of birds, as the aerodynamicmechanisms of insects and birds are not fundamentally different.

Wing twist can however be observed on some birds in slow speed flight (e. g. Rosénet al., 2004). Recently, two studies managed to visualize the flow directly around theflapping wings of slowly flying birds (Muijres et al., 2012c; Chang et al., 2013). ProminentLEVs were found, and it seems that wing twist in slow-speed flight is not used to avoidthe development of LEVs, but rather to modulate their size and stability and to directthe resultant force. Maybe, the application of wing twist is generally not used to decreaseαeff at the wing tip, but to increase αeff at the inner part of the wing. This would resultin high angles of attack and high aerodynamic forces over the full wing – however at thecost of efficiency. Further flow visualizations of the fluid directly around the wings ofbirds flying at several speeds are highly desirable to validate the results and to get furthervaluable information on the control of flow separation in birds.

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The effects of wing twist

CONCLUSIONS

Wing twist was assumed to be essential in the flapping flight of birds in order to keep theeffective angle of attack sufficiently low. It was shown that this is not strictly necessary,and that reducing the effective angle of attack at the wing tip reduces the aerodynamicforce – in analogy to the flapping flight of insects. It is likely, that such a reduction in thepeak aerodynamic force is undesirable in many situations in avian flight. In slow flighthowever, the purpose of wing twist might not be the reduction of the effective angle ofattack at the wing tip, but the increase of αeff at the wing base, making the whole wingoperate at high effective angles of attack. The mechanical flight efficiency (related toLcirc/Dind) as well as the efficiency of lift generation (related to span efficiency) degradewhen αeff is increased – similar to a wing in purely steady conditions. But even if theaerodynamic efficiency significantly drops, the overall fitness of a bird is supposed toincrease due to the ability to generate larger forces.

The ability to modify the force coefficients of the wings via wing twist and the resultingchanges inαeff will largely enhance the flight envelope, making flapping wing locomotionvery attractive. From the engineering point of view, the cruising flight of flapping-wingvertebrates might be of secondary interest, because existing fixed wing airplanes alreadyoutperform birds in terms of the energetic cost of transporting a unit of weight over aunit of distance (cost-of-transport, Tucker, 1970). In contrast, the manoeuvrability andversatility of avian flapping flight is yet totally unmatched by any technical applicationand it is an important source of inspiration for researchers and engineers today and inthe future.

125